Covalent Benzenesulfonic Functionalization of a Graphene Nanopore for Enhanced and Selective Proton Transport

A fundamental understanding of proton transport through graphene nanopores, defects, and vacancies is essential for advancing two-dimensional proton exchange membranes (PEMs). This study employs ReaxFF molecular dynamics, metadynamics, and density functional theory to investigate the enhanced proton transport through a graphene nanopore. Covalently functionalizing the nanopore with a benzenesulfonic group yields consistent improvements in proton permeability, with a lower activation barrier (≈0.15 eV) and increased proton selectivity over sodium cations. The benzenesulfonic functionality acts as a dynamic proton shuttle, establishing a favorable hydrogen-bonding network and an efficient proton transport channel. The model reveals an optimal balance between proton permeability and selectivity, which is essential for effective proton exchange membranes. Notably, the benzenesulfonic-functionalized graphene nanopore system achieves a theoretically estimated proton diffusion coefficient comparable to or higher than the current state-of-the-art PEM, Nafion. Ergo, the benzenesulfonic functionalization of graphene nanopores, firmly holds promise for future graphene-based membrane development in energy conversion devices.

S1. Supporting Computational Methods S1.1 A further validation of the ReaxFF force field via pKa estimation of the benzenesulfonic acid.In this section, we had further validated the ReaxFF CHONSMgPNaTiClFKLi.ffforce field 1,2 for describing the benzenesulfonic functionality.This validation was carried out using unbiased molecular dynamics to extrapolate the reaction free energy for the deprotonation process and, subsequently, determine the acidity constant, pKa. 3,4For this validation, we employed a simulation box with dimensions of 12.91 Å × 12.91 Å × 12.91 Å, containing a benzenesulfonic acid molecule solvated by 62 water molecules (Figure S1a).Three unbiased ReaxFF-MD NVT equilibrations of 1.5 ns each were conducted using the LAMMPS software, 5 with the CSVR thermostat 6 maintaining a temperature of 300 K.The simulations utilized a time step of 0.25 fs and a damping constant of 100 fs for temperature control.During these simulations, a Collective Variable 4 (CV4) was employed. 4e CV4 is defined as the distance,  =  −  , between the hydronium and the sulfur of the benzenesulfonic acid.The  represents the position of the sulfur atom of the benzenesulfonic acid, while  , denotes the position of the hydronium ion, computed as a weighted sum over all positions,  , of oxygen atoms capable of binding the proton: with the weights, The position  is thus obtained as an exponentially weighted average of all proton acceptor positions, which are the water and oxygen atoms of the benzenesulfonic acid.With a positive number for the parameter , the weighting selects the oxygen(s) with highest coordination number with respect to the H-atoms,  .Here  , is the default number of hydrogens bonded to atom i (without the proton), which is 2 for a water oxygen and 0 for the benzenesulfonic acid oxygen atoms.The coordination number  =  is computed with: where  = O and  = H.The estimated averaged free energy profile and evolution of the CV4 are shown in Figure S1b and Figure S2, respectively.The free energy profile along the path shows three states (Figure S1b).The first minimum at 1.7 Å corresponds to the proton bonded to the benzenesulfonic functionality; the top of the energy barrier corresponds to the sharing of the proton with the first water solvation shell, CV4 = 2.38 Å; the second minimum is associated to the proton solvated in the first water hydration shell (contact ion pair), 4 CV4 ≈ 3.4 Å; and finally the proton diffuses in the water bulk CV4 ≥ 4 Å.The pKa of the benzenesulfonic acid can be obtained from the reaction free energy difference using the following equation: 4 p = Δ   ln (10)  = −1.52 The obtained pKa of −1.52 is in a reasonable agreement with the experimental value of −2.50 reported in the Supporting Information, section S2.4 Table S1. 7This analysis provided a further validation of the ReaxFF force field CHONSMgPNaTiClFKLi.ffforce field 1,2 in order to be used for estimating the energetics and dynamics of a graphene nanopore functionalized with benzenesulfonic group in water environment.
S1.2 Water density ratio in the graphene nanopore relative to the bulk.The water densities for the hydrogenated and covalently functionalized graphene nanopore systems are computed employing the MDAnalysis package. 8,9The calculation of the water density involved the utilization of the oxygen atom positions from each ReaxFF-MD metadynamics simulation for each of the hydrogenated and functionalized graphene nanopore systems (Figure 4a,c,e).From the water density data, the water density ratios of each system reported in sections 3.1 and 3.4, averaged over the corresponding three ReaxFF-MD metadynamics simulations, were extracted relative to the water bulk value (ρwater ~ 1.0 g cm −3 ) , normalized within a 3 Å thick layer respect to the center of mass of the graphene layer (Figure S7, Figure S20).

S2 Supporting Results
S2.1 Free energy profiles along the Collective Variable CV1 of the flexible and rigid hydrogenated graphene nanopore systems.The free energy associated with proton transport for both a freely moving, i.e. flexible, and a constrained, i.e. rigid hydrogenated graphene nanopores, was determined to assess the role of graphene sheet undulations.The weak symmetry breaking of the free energy profiles around the nanopore center reflects finite sampling, and it is slightly more prominent when the graphene is flexible.Figure S3 suggests that the motion of the graphene sheet is likely to result in an increase in the energy barrier for proton transport across the hydrogenated graphene nanopore when compared to the rigid case, with an estimated change in free energy of approximately ΔΔF ≈ 10.7 ± 2.8 kJ mol −1 .S2.2 Selectivity of proton transport through a hydrogenated graphene nanopore over sodium cation.To analyze the proton selectivity, the density of the water shell around the graphene nanopore was visualized, see Figure S6a,c.1][12] On the other hand, water can penetrate the hydrogenated graphene nanopore as seen in Figure S6b,d showing the atomistic positions of hydrogen and oxygen.From the water density data, the water density ratio reported in section 3.1 were extracted relative to the water bulk value (ρwater ~ 1.0 g cm −3 ) averaged within a 3 Å thick layer respect to the center of mass of the graphene layer (Figure S7).In Figure S8, Figure S9, and Figure S10 we depict the analysis of the selectivity of the flexible hydrogenated graphene nanopore with respect to a sodium cation (Na + ), as discussed in Section 3.1.

S16
S2. 5 The benzenesulfonic functional group as a shuttle in the proton transport process: energetic and dynamics.b, Representative configuration for the benzenesulfonic functionalized graphene nanopore system.The benzenesulfonic group, the sodium cation (Na + ) and the water molecules involved in its first solvation shell are represented by balls and sticks: carbon, oxygen, sulfur, hydrogen, and the sodium cation are colored in grey, red, yellow, white, and light grey, respectively.

The
CV4 is not highly sensitive to the exact values of the switching function, and the same parameters of the switching function are effective across the entire pKa spectrum.The following parameters values were used:  = 12,  = 24,  = 8, and  = 1.3 Å.

Figure
Figure S1.a, Representative configuration for the benzenesulfonic acid solvated in water.The benzenesulfonic acid is represented by balls and sticks: carbon, oxygen, sulfur, and hydrogen are colored in grey, red, yellow, and white, respectively; water molecules are represented by sticks.b, Free energy profile (kcal mol −1 ) for proton dissociation reaction along the Collective Variable CV4 (Å) averaged over 1.5 ns each of three independent simulations for benzenesulfonic acid solvated in water.

Figure S2 .
Figure S2.Time evolution of the CV4 (Å) for the simulation of benzenesulfonic acid in water solvation.

Figure S3 .
Figure S3.Free energy profiles (kJ mol −1 ) of proton transport through flexible (solid line) and rigid (dashed line) hydrogenated graphene nanopore systems along the Collective Variable CV1 (Å) averaged over 1.0 ns each of the three final independent simulations.

Figure S4 .
Figure S4.Free energy convergence (kJ mol −1 ) of proton transport through flexible (top) and rigid (bottom) hydrogenated graphene nanopore systems along CV1 (Å).To assess the convergence of a metadynamics simulations, each free energy profile is extracted after 0.25 ns (violet line), 0.5 ns (cyan line), 0.75 ns (green line) and 1.0 ns (orange line) of each simulation, with a deposition stride every 25 fs, and the global minimum is set to zero in all profiles.

Figure S5 .
Figure S5.Time evolution of the Collective Variable CV1 (Å) for each of the three final independent simulations of the flexible (top) and rigid (bottom) hydrogenated graphene nanopore systems, respectively.

Figure S6 .
Figure S6.Time-averaged water density extracted from one representative ReaxFF-MD metadynamics simulation for the a, flexible and c, rigid hydrogenated graphene nanopore systems in the x-z plane, averaged along the y-axis.The thickness

Figure S7 .
Figure S7.Density profiles (g/mL) of water along the relative distance to the graphene (Å) for the a, flexible and b, rigid hydrogenated graphene nanopore systems.The void zone of ~ 5 Å and the region for the water density ratio estimations have been highlighted.The graphene layer is represented with a gray dashed line by way of illustration.

Figure S8 .
Figure S8.Time evolution of the Collective Variable CV1 * (Å) for the three independent simulations of the flexible hydrogenated graphene nanopore system with a sodium cation in the water bulk.

Figure
Figure S9.a, Free energy profile (kJ mol −1 ) of the sodium cation transport through a flexible hydrogenated graphene nanopore system along the Collective Variable CV1 * (Å) averaged over 2.0 ns each of the three final independent simulations.b, Representative configuration for the flexible hydrogenated graphene nanopore system.The sodium cation (Na + ) and the water molecules involved in its first solvation shell are represented by balls and sticks: carbon, oxygen, hydrogen, and the sodium cation are colored in grey, red, white, and light grey, respectively.

Figure S10 .S2. 3
Figure S10.Free energy convergence (kJ mol −1 ) of sodium cation transport through the flexible hydrogenated graphene nanopore system along CV1 * (Å).To assess the convergence of a metadynamics simulations, each free energy profile is extracted after 0.25 ns (violet line), 0.5 ns (cyan line), 0.75 ns (green line), 1.0 ns (orange line), 1.25 ns (blue line), 1.5 ns (red line), 1.75 ns (dark green line) and 2.0 ns (black line) of each simulation, with a deposition stride every 25 fs, and the global minimum is set to zero in all profiles.

Figure
Figure S13.a, Schematic representation of the graphene nanopore covalently functionalized with benzenesulfonic and the Collective Variable CV3, in red, defined as the difference between z-components of the center of mass of the oxygens (in green) of the benzenesulfonic functionality and the center of mass of the graphene nanopore calculated from the position of the carbon atoms highlighted in purple.b, Free energy profile (kJ mol −1 ) along the Collective Variable CV3 (Å) averaged over 2.0 ns each of the three final independent simulations for the benzenesulfonic (Ph-SO3H) covalent functionalized graphene nanopore system.

Figure S14 .
Figure S14.Free energy convergence (kJ mol −1 ) of proton transport through benzenesulfonic (Ph-SO3H) covalently functionalized graphene nanopore system along CV3 (Å).To assess the convergence of a metadynamics simulations, each free energy profile is extracted after 0.25 ns (violet line), 0.5 ns (cyan line), 0.75 ns (green line), 1.0 ns (orange line), 1.25 ns (blue line), 1.5 ns (red line), 1.75 ns (dark green line) and 2.0 ns (black line) of each simulation, with a deposition stride every 25 fs, and the global minimum is set to zero in all profiles.

Figure S15 .
Figure S15.Time evolution of the Collective Variable CV3 (Å) for the three final independent simulations of the benzenesulfonic (Ph-SO3H -blue points) covalent functionalized graphene nanopore system.

Figure S17 .
Figure S17.Separate free energy profiles (kJ mol −1 ) along the Collective Variables CV1 (Å) and CV2 (Å) in blue and red lines, respectively, over 1.0 ns each, of the three final independent simulations of the graphene nanopore functionalized with a benzenesulfonic (Ph-SO3H) group.

Figure S19 .
Figure S19.Time evolution of the Collective Variables CV1 (Å) and CV2 (Å) in blue and red lines, respectively, for each of the three final independent simulations for graphene nanopore functionalized with a benzenesulfonic (Ph-SO3H) group.

Figure S20 .
Figure S20.Density profiles (g/mL) of water along the relative distance to the graphene (Å) for the a, benzenesulfonic (Ph-SO3H), b, benzoic (Ph-COOH), and c, phenol (Ph-OH) covalent functionalized graphene nanopore systems.The void zone of ~ 5 Å and the region for the water density ratio estimations have been highlighted.The graphene layer is represented with a gray dashed line by way of illustration.

Figure
Figure S21.a, Time evolution of the Collective Variable CV1 * (Å) for the three final independent simulations of the benzenesulfonic covalent functionalized (Ph-SO3H) graphene nanopore system with a sodium cation in the water bulk.
TableS1for the vacuum and COSMO simulations show an increase in affinity from the benzenesulfonic acid to the phenol group.This result is in line with the experimentally determined pKa.7 TableS1.DFT Proton Affinity estimations (kJ mol −1 ) for benzenesulfonic acid, benzoic acid, and phenol molecular systems, in vacuum and water continuum model solvent (COSMO), and the corresponding experimental pKa values.